Indeterminate forms come from limits. It’s not the question you asked, and I think this answer was a little off the mark because of it. For the sake of shared knowledge, I will explain anyways:
When looking at a limit, it’s important to note that you aren’t working with zero (or infinity, or any number you are studying the limit of), what you are working with are numbers approaching the limit. For example, for (x+1)/(x), the expression has no equivalent value at x=0, as 1/0 does not exist. We can see why if we use the limit as x approaches zero. The numerator will approach 1, and the denominator approaches 0. The numerator has little impact on the value of the expression, but the denominator… dominates the value, for the pun. And, while we can’t evaluate at 0, we can put really small numbers in there and see what happens- and what happens is the expression becomes incredibly large. I’m sure that if you don’t see where this is going, you can go to Desmos or some other graphing calculator and try it for yourself.
As far as the indeterminate form- 0/0 is always undefined, at least in most mathematics. However, if you were to look at equations :
y = x/x
y= x^2^/x
y= x/x^2^
you’ll see the curves behaving differently around x=0. The first makes 0/0 look like 1, the second makes 0/0 look like 0, and the last will make 0/0 look like infinity*. Once again, note, however: 0/0 does not exist, and there is discontinuity on all of these curves at x=0.
*Edit: or negative infinity, I forgot that this limit doesn’t exist. Even though the limit doesn’t exist, it is still a useful example.
It’s a calculus thing. We can only give the expression a value if we know the functions giving us a zero value that are being devided. For example if we were dividing the function (X) by the function (X^2) at zero our we would get infinity (Wikipedia has a pretty good page on indeterminate forms).
You could also think of it like multiplying both the numerator and denominator of a fraction by 0. This should preserve the fractions value, but multiplying by 0 essentially erases both values so we can no longer know what the fraction equals unless we know how both values came to be 0.
“I suppose the surprising thing is why the atmosphere doesn’t all fall immediately to the Earth’s surface to form a thin dense layer of air molecules.
The reason this doesn’t happen is that air molecules are all whizzing around at surprisingly high speeds - typically hundreds of metres per second depending on the temperature.
The air molecules bash into each other and knock each other around, and the air molecules near the ground bash into the air molecules above them and stop them falling down.”
If it were zero then all the air would indeed just fall down to the ground (actually, this is a simplification I’ll address later).
As you increase the temperature the atoms of the ground will start to wiggle more and they’ll start to kick the air molecules giving them non-zero average height.
So the atmosphere would move a little off the ground. The bigger the temperature is the higher the atmosphere will reach.
Note: there are number of assumptions above that simplify the picture. They are not that important but I want to provide a complete picture:
1, Even at the zero temperature the molecules would wiggle a little because of quantum mechanics
2, The atmosphere would freeze at some point (like 50K) so under that temperature it would just lie on the ground
3, I assumed that the ground and the atmosphere have the same temperature because they are in the thermal equilibrium; in reality their temperatures can differ a little because of additional slow heat-transfer processes.”
This is the way! It helps me to imagine what would it look like if the atmosphere consisted of a single nitrogen molecule. You place it on the ground but the ground has temperature (is warm) so your one molecule gets launched up into the vacuum on a parabolic trajectory at 500 m/son average. If it launched at 45° it would reach 6km up and fall down, at 90° - 12km up - and that’s on average. Some would get launched faster and higher (following the long tail of the Boltzmann distribution), and hydrogen and helium even faster still because they are lighter. A few hydrogen molecules would be launched at speed above 11km/s, which is above Earth’s escape velocity, so they would escape and never fall down.
When you have many air molecules, they hit each other on the way up (and down), but because their collisions must be perfectly elastic, mathematically it works out that the overall velocities are preserved. So when your one nitrogen molecule gets launched up but on its way hits another identical molecule, you can think of them equivalently as passing through each other without colliding at all. (Yes, mathematically they can also scatter in some other random directions, but the important part is that your original molecule is equally likely to be boosted further upwards as opposed to impeded.)
The end result is that majority of the atmosphere stays below 12km, density goes down as you go up though never quite reaching zero, and hydrogen and helium continuously escape to space to the point none are left.
There’s probably even a small handful of foods doing most of it, like white bread, sweetened cereal, soda. However maybe the trick is to require livable wages for all those fast food workers and hope there’s some truth to franchisee claims that it becomes an unsustainable business
There’s probably even a small handful of foods doing most of it
Sadly, no. This is a plague that goes across the board when it comes to processed foods. While some are worse offenders than others, some even healthy appearing foods (cereals, smoothies) are horrible dangers for your health.
My wife stopped eating out for nearly every meal recently and started cooking 90% of what we consume, best decision we’ve made for our health and our pocketbook. Food is processed less and we can control what’s in it, and in turn we’re both almost back to our college weight. Turning fresh produce and protein into a meal is where it’s at.
A Dyson swarm is basically just a huge number solar collectors orbiting the sun. Humanity could put some individual collectors in space if we wanted to, but we don’t have anywhere near enough resources to make a full swarm.
Near-relativistic spacecraft are conceivably possible and are not too far beyond what’s possible with current technology (though would still require significant advancements). The catch is that they would be very tiny and we would have to send a stream of them to their destination.
Retinal projectors are currently under development, and advanced ones could in principle be higher quality than current VR headsets while having a very small form-factor. Optical metamaterials such as metalenses would be very useful for this, particularly if they could be designed to work at all three RGB wavelengths simultaneously (not easy).
I think you’d be better served to ask about the average volume of a cloud (if that makes sense, given how diverse they come). Because the mass is pretty much exactly volume multiplied with density. And the average density of clouds is pretty much exactly that of the surrounding air at the given altitude (because otherwise the cloud would not float, but either rise up or sink down). And the density of air at any given altitude is given by the Barometric formula. If you take a kubik kilometer of cloud (honestly, I have no idea how big clouds are), it would have a mass of approximately 364 thousand tons at 11 km above sea level, 88 thousand tons at 20 kilometers above sea level, 860 tons at 51 km above sea level and ca. 64 tons at 71km above sea level. But “regular” clouds only go up to ca. 20km, and 95% of the clouds you see are probably below 8km. Unfortunately the quoted wikipedia page has no entry for the barometric pressure at that altitude and I am too lazy to try and calculate it right now ;)
Hi! Googling this question reveals the answer is yes, it does result in fusion.
As far as the output, according to this top result paper, that depends heavily on the size of the black hole, the size and speed of the accretion disk, and the medium from which the black hole is drawing (like a white dwarf vs interstellar gas).
From what I can make out–and I have no background–the author maps out results as high in weight as nickel.
I won’t lie, if you don’t close the lid and I know it, we’ve nothing to ever speak about because that’s disgusting and please stay away from me, I don’t want your toilet aura near me. 💀
No, as both gravitational and em waves travel at the speed of light, the “we’re all screwed” things we could ever observe would only ever be slowed-down/distorted (by things that could even do such a thing like a black hole) as they approach us.
So it’d be a happy little surprise (short of worm-holes or tachyons existing), on year 26,000.
That’s not too say life as we know it would end immediately. We might make it generations before the real chaos affected us on earth. On a smaller scale, if the sun blipped out of existence, sure, we’re 💯 doomed and we’d know it after 8.3 minutes, but some of us might make it a solid week before all life on earth was expunged 😅.
The wavelength of a photon isn’t intrinsic to the photon itself—it depends on the observer’s inertial frame. The Hubble redshift occurs because expansion affects the velocity of observers relative to the photons’ original frame, not because it affects photons directly.
So I might perceive a stream of photons as radio waves from our current inertial frame, but if I were on a ship approaching the speed of light head-on towards that same stream of photons, I might perceive them as visible light? Or ultraviolet or gamma ray.
Wait… that doesn’t sound right, for some reason.
It took billions of years of universe-stretching for the CMB to redshift to microwave, I don’t think me pushing pedal to the metal for a few seconds on a rocket ship is going to counteract all those years.
Then also those CMB photons are more diffuse, spread out.
Doesn’t Coloumb play into this?
It took billions of years of universe-stretching for the CMB to redshift to microwave, I don’t think me pushing pedal to the metal for a few seconds on a rocket ship is going to counteract all those years.
We do see shifts in the CMBR due to local velocity changes though—for instance, we can tell that the sun is moving at about 370 km/s relative to the CMBR frame due to its radial movement through the galaxy and the motion of the galaxy itself through space.
Sure, that I do get, it’s just that I’m guessing my local movement at the speed of light for a moment can blueshift a radiowave or microwave photon only a fraction of what is needed to get it into the visible light spectrum, surely never all the way to ultraviolet or gamma rays.
Except for the young and the pregnant, we’re all wearing red shirts out here. In nature, most living things are highly disposable.
It’s an uncomfortable truth that is also weirdly comfortable at times. As far as nature is concerned I’m a spear carrier who should have been dead a long time ago…this is all gravy, baby!
According to this Stack Exchange answer, glass reflects around 4-100% of the UV in sunlight depending on the angle of incidence. So you could probably get a sunburn if the angle is low enough (like if the Sun is almost directly overhead and reflecting off a vertical window).
I thought you can’t digest more than a little bit of blood because of the amount of iron in it. And you’re likely to start vomiting.
And if you lose too much blood, it’ll kill you much more quickly anyways.
And of course if you lose blood and have to replenish it… That takes a good amount of extra energy to produce all the blood cells etc. And digestion also costs energy.
One more thing: adding hot water, which evaporates faster, will probably increase vapor pressure in the environment, slowing down, or even stopping evaporation.
The Hubble constant is an interesting one – it isn’t actually a constant, but if you reframe it as a partial differential equation, the law is very predictable. en.wikipedia.org/wiki/Hubble's_law#Time-dependenc… – so in many ways, it’s just a misnamed phenomenon, and shouldn’t be called a constant at all.
Any place in the universe beyond which we’ve directly sent probes – is assumed to be like those parts we already know. Part of the reason we make this assumption is that: main sequence stars appear to behave identically across vast reaches of space and time. Thus we assume that physics hasn’t changed significantly (at least within the period of time where main sequence stars exist). Because if the physics was different, the stars would be different (spectra, lifecycles, etc.).
I’ll present a tickler I learned in cosmology decades ago, for hand-waving.
Run the big bang backwards – imagine all of the matter and energy of the universe collapsing to a single point. Which point is at the centre? They all are. Run time forwards again and all the points expand outwards from each other, but which point was at the centre that you can use to reference the centre of the universe against? They all were. Thus, I am the centre of the universe. And so are you ;)
This made my brain melt until I learned to visualize this using lower dimensional surfaces (like Riemann spheres). Imagine a beach ball being inflated. It is a two dimensional surface. You’re an ant on the beach ball and all the other points are getting further away, but it’s happening in a uniform way. (The Hubble parameter is something like the rate at which air is added to the beach ball.) Now, run this beach ball backwards through time – it shrinks and shrinks until it becomes a single point, where all points overlap – every point is the centre of the beach ball universe. Run this forward in time again and ask: which point on the surface of the beach ball is the centre of this two-dimensional universe? And the answer is “all of them” and the universe should be uniform in its expansion properties.
I remember reading an article (can’t find it right now) from a PHD dropout who was doing research in string theory. One of reasons he dropped out is his frustration at how abstract and disconnected from reality his work was. His advisor (and his colleagues) didn’t have that problem, because to him, the math behind string theory was an ends in itself. There’s beauty in math, regardless of whether it has any practical application. If string theory turns out to be an accurate model of reality, then that would be a nice bonus, but that’s not why his advisor studied it.
So to answer your question, if we somehow reach the point where everything that can be feasibly discovered has been discovered, then theoretical scientists would make up their own models and study those.
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